Very. Almost all of NYC is very, very safe.
Yes - that's where the celebs live.
Yes, that's where all the rich people live. DeNiro, Jay-Z.
However in general don't take New York as the safest place in the world
Tribeca is full of rich people, who will ignore you on the street. You will be quite safe, but keep your eyes on your valuables as you would anywhere. Don't leave things unwatched when you shop, or eat, and keep your money secure. The worst you will face in Tribeca is pickpockets on the crowded streets near Broadway, and the hawkers in China-town and along Canal selling you, "DVDs, Cds, Rolexes and designer bags!"
TriBeCa is busy with restaurants and clubs. Stay on the main streets with lots of people milling around. Avoid dark alleys. This is the advice I got from my folks when I was growing up in NYC. I lived in a beautiful residential neighborhood with lots of doormen, and I always felt safe. When I went downtown, I just stayed with the crowd. In decades of exploring the city day and night, I never had an incident.
Yes, Tribeca is a safe area. Most of New York City is safe. This includes most of the Bronx and Brooklyn, too.
New York is big (as in enormous), loud, fast paced and intense. Many people confuse this with dangerous. To be sure, there are places in New York that aren't as safe as others, but that's true of every city and town on earth.
Just be aware of your surroundings. If something makes you uneasy, hail a cab or leave that area.
Trust your instincts. If it doesn't seem right, then it's not right for you. It doesn't matter what others think; you're the one you have to answer to.
I grew up the New York city area in the 1970s, when it was openly called "Fear City": A dirty and dangerous place. Well, that was over thirty years ago. In that time the city and many of it's outer boroughs have come back in a big way. New immigrants, new money and a new spirit have completely renovated the image of the terrible city portrayed in so many low budget action films of the 1980s.
Since then I've been to many places in the United State that I feel are far more frightening than New York was at it's worst. Many of these are in middle America, and some even in rural New England.
As for the citizens of New York, they're OK. Oh sure, they're blunt, and in many cases have rather intense personalities, but largely there good folks. Give 'em a chance. They just may surprise you!